Proposition 19.7.2 (Cocartesian semiring construction). The functor \[ \Cat _{\infty }^{\mathrm {coprod}}\longrightarrow \CMon (\Cat _{\infty }), \qquad C\longmapsto (C,\amalg ), \] has a canonical lax symmetric monoidal refinement, where the source carries \(\otimes _{\mathrm {coprod}}\) and the target carries the symmetric monoidal structure of Proposition 16.4.1.
Proof. Write \[ U\colon \Cat _{\infty }^{\mathrm {coprod}}\longrightarrow \Cat _{\infty } \] for the forgetful functor. The multimorphism operad defining \(\otimes _{\mathrm {coprod}}\) is, by construction in Section 22.4, a suboperad of the cartesian multimorphism operad of \(\Cat _{\infty }\). This inclusion gives \(U\) a canonical lax symmetric monoidal structure.
By Lemma 19.7.1, the multimorphism operad of \(\Cat _{\infty }^{\mathrm {coprod}}\) is semiadditive. Moreover, \(U\) preserves finite products, since they are computed by cartesian products of \(\infty \)-categories on both sides. The adjunction of Theorem 18.4.12 therefore gives a unique lift
By the definition of \(\oCMon \) and the Day convolution structure of Proposition 16.4.1, the upper-right operad is naturally equivalent to the multimorphism operad of \(\CMon (\Cat _{\infty })\): \[ \oCMon \bigl (\Mm _{(\Cat _{\infty },\times )}\bigr ) \simeq \Mm _{\CMon (\Cat _{\infty })}. \] The lift consequently determines a lax symmetric monoidal functor from \(\Cat _{\infty }^{\mathrm {coprod}}\) to \(\CMon (\Cat _{\infty })\). On an object \(C\), it gives the commutative monoid structure induced by the biproducts in \(\Cat _{\infty }^{\mathrm {coprod}}\). Its unit selects the initial object \(0_C\), while its addition is the codiagonal \[ C\times C\simeq C\oplus C\longrightarrow C. \] Under the description of the coproduct in Lemma 19.7.1, this codiagonal is the coproduct functor \(\amalg \colon C\times C\to C\). Thus the underlying functor sends \(C\) to \((C,\amalg )\), as claimed. □
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